(************** Content-type: application/mathematica **************

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Notebook[{

Cell[CellGroupData[{
Cell["Cassinian Oval", "Title"],

Cell[CellGroupData[{

Cell["Note", "Subsubsection",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[{
  "This Mathematica notebook is from the WWW project A Visual Dictionary of \
Special Plane Curves. Copyright \[Copyright] 1995-2002 by Xah Lee. (",
  ButtonBox["http://xahlee.org/",
    ButtonData:>{
      URL[ "http://xahlee.org/"], None},
    ButtonStyle->"Hyperlink"],
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  AspectRatioFixed->True]
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Cell[CellGroupData[{

Cell["Packages and Initializations", "Subsubsection",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[BoxData[{
    \(Needs["\<Graphics`ImplicitPlot`\>"]\), "\n", 
    \( (*MathSource\ \(Item : 0200 - 529\)*) \)}], "Input",
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Cell[BoxData[
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Cell[CellGroupData[{

Cell["Cassinian Oval as a torus cut, proof", "Subtitle"],

Cell[CellGroupData[{

Cell["problem", "Subsection"],

Cell["\<\
prove that cassinian oval Sqrt[(x - a)^2 + y^2]*Sqrt[(x + a)^2 + \
y^2]==b^2 is the intersection of a plane and a torus.\
\>", "Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell["solution", "Subsection"],

Cell["\<\
The formula of torus with tube radius c and its center to torus \
center d is\
\>", "Text"],

Cell[CellGroupData[{

Cell["(Sqrt[x^2 + y^2] - d)^2 + z^2 == c^2", "Input"],

Cell[BoxData[
    \(\((\(-d\) + \@\(x\^2 + y\^2\))\)\^2 + z\^2 == c\^2\)], "Output"]
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Cell[CellGroupData[{

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Cell["\<\
we want to find the intersection of this torus with a plane \
x==c\
\>", "Text"],

Cell["eliminate x by replacing it with a we have:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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square both sides. (we know that the expression under the square \
root is positive.\
\>", "Text"],

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Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell["\<\
We want to show that the above equation has the Cassinian oval form \
defined as
Sqrt[(x - a)^2 + y^2]*Sqrt[(x + a)^2 + y^2] == b^2
(possibly up to scale, rotation, and translation.)\
\>", "Text"],

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Cell[CellGroupData[{

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square both sides (the radicands are positive in our context since \
they are distances), regroup, and expand we have\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell["change the variable in the cut torus", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell["\<\
We see that the two are similar. Now we just let d=a and \
c=b^2/(Sqrt[4]*a)\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell["end of proof.", "Text"]
}, Open  ]]
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